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What does checking behind flops do to counter the strategy?

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It increases the button's positional advantage.

The button has the option of selectively checking behind and taking the free-card or betting when he likes the flop, knowing full-well that Big Blind's average hand is stronger than it would otherwise be if he was 3-betting pre-flop.

Big Blind's whole game-plan is to forfeit some value (an extra SB) by not ever raising pre-flop, with the expectation that he will get that value back on the flop with a check-raise. And he prefers to do it on the flop because he'll have gained flop information. All Button has to do is not go along with Small Blind's grand plan and check behind him more liberally than he usually would.